How to fit the eyes in character?
To make the eyes fit properly in your character's head there are a couple of things to keep in mind.


Eyeball size: If the eyes are incorrectly sized they will not fit right no matter what you do.
Eye position: Make sure the eyes are in the correct location on all axes. One thing to try is to move the eye just slightly above center on the z axis (so that the center of the eye socket opening is very slightly below the center of the eye sphere).
Eye socket geometry: If you look at references for eyes you can get an idea of how the eye socket is shaped and how the eyelid is around the eye. Many people often forget to include the tear duct at the inner corner of the eye, and that can cause a lot of issues.


The main thing to keep (which encompasses all of what I said above) is how eyes look in real life. Find some reference images of a character similar to what you are going for and keep them handy to refer to.